Much Ado about Zero

American Mathematical Society "All for Nought" Feature Column. Used with permission.Chatur-bhuja temple in Gwalior (south of Agra), India, and part of an inscribed stone tablet dated to 876 CE. The blue circle shows the number 270 (as in, "...the whole town gave to the temple...a piece of land...270 hastas in length...").

American Mathematical Society "All for Nought" Feature Column. Used with permission.Notice how similar the glyphs (reproduced in red) are to our 2, 7 and 0.

American Mathematical Society "All for Nought" Feature Column. Used with permission.

No simpler concept exists in arithmetic than the "number line," which stretches from minus infinity through (-3, -2, -1, 0, 1, 2, 3) to infinity, and contains every "regular" number. Even or odd, positive or negative, prime (divisible only by itself and one) or composite, rational (2/3) or irrational (?, ?2) -- the diagram says that for every quantifiable number, as West Side Story puts it, "there's a place for us."

It wasn't always so. The brightest mathematicians in Sumeria, Babylon, Greece, Rome and India took millenia to establish the simple elegance of the number line. In particular, the concept of zero as a "regular" number became commonplace only in the last few hundred years.

The Babylonians, who used a base-60 number system, left gaps on their clay tablets as "placeholders" (in the number 206, the zero is a placeholder indicating no tens). The lack of an actual symbol means that, for instance, Babylonian 3 and 180 -- i.e. 3 x 60 in base-60 -- look the same, distinguished only by context. (It would be like writing 5 in our system, meaning 5, or 50 or 500.)

• Similarly, the Inca system of keeping track of numbers with knotted cords, or quipus, lacked an actual symbol for zero; the absence of a knot indicated zero.

• The earliest unequivocal use of a glyph, or symbol, for zero -- that is, treating zero as a number rather than an absence of number -- is a stone inscription in a temple at Gwalior, India, dated 876 CE (photo). And indeed it looks very much like the circle we now use to indicate zero. (Indian Jain mathematicians had been using zero hundreds of years earlier, but in words, not as a single glyph.)

• Even the great Leonardo of Pisa (1170-1250), aka Fibonnaci, who introduced the decimal system and Arabic numerals to the west, couldn't get his head around the idea of zero as a regular-guy, beer-swilling number. He wrote, "The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written." By "sign," he was thinking of zero not as a number, but as an operator -- like a plus or minus sign.

• The number 0 is neither positive nor negative, neither prime nor composite -- it cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed by multiplying prime numbers.

"... the loneliest number that you'll ever do" is zero, whatever Three Dog Night sez.

Barry Evans (barryevans9@yahoo.com) worrys about the difference between "a big fat zero" and "a small thin zero." His books are at Eureka Books.